# Biholomorphic mapping

holomorphic isomorphism, holomorphism, pseudo-conformal mapping

A generalization of the concept of a univalent conformal mapping to the case of several complex variables. A holomorphic mapping of a domain $D\subset\mathbf C^n$ onto a domain $D'\subset\mathbf C^n$ is said to be a biholomorphic mapping if it is one-to-one. A biholomorphic mapping is non-degenerate in $D$; its inverse mapping is also a biholomorphic mapping.

A domain of holomorphy is mapped into a domain of holomorphy under a biholomorphic mapping; holomorphic, pluriharmonic and plurisubharmonic functions are also invariant under a biholomorphic mapping. If $n>1$, biholomorphic mappings are not conformal (except for a number of linear mappings) and the Riemann theorem is invalid for biholomorphic mappings (e.g. a ball and a polydisc in $\mathbf C^2$ cannot be biholomorphically mapped onto each other). A biholomorphic mapping of a domain $D$ onto itself is said to be a (holomorphic) automorphism; if $n>1$, there exist simply-connected domains without automorphisms other than the identity mapping.

#### References

 [1] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)

Concerning boundary behaviour of biholomorphic mappings the following results have been obtained. C. Fefferman's theorem: A biholomorphic mapping between strongly pseudo-convex domains with $C^\infty$-smooth boundary extends $C^\infty$-smoothly to a diffeomorphism between the closures of the domains, see [a3]. The same result holds if the domains are only pseudo-convex and one of them satisfies condition $R$ for the Bergman projection, see [a2]. For strongly pseudo-convex domains with $C^k$-boundary, $k>2$, $C^{k-1-\epsilon}$ extendability was obtained ($\epsilon>0$ if $k=2,3,\dots,$ $\epsilon=0$ otherwise) by L. Lempert and by S. Pinčuk. For (weakly) pseudo-convex domains with real-analytic boundary one has even holomorphic extension to a neighbourhood of the closure, see [a1]. Similar results were obtained for proper holomorphic mappings.
A biholomorphic mapping $f$ is proper (i.e. the pre-image of a compact set is compact), since $f^{-1}$ is continuous. Riemann's theorem does not hold in the following sense: There is no proper holomorphic mapping from the polydisc in $\mathbf C^n$ onto the ball in $\mathbf C^m$ for any $n,m>1$, cf. [a4]. Thus, function theory in $\mathbf C^n$, $n\geq1$, is strongly related to the domain of definition of the functions. For function theory in the (unit) ball of $\mathbf C^n$ see [a5]; for function theory in polydiscs see [a6]. For entire holomorphic mappings and their value distribution see [a7].
 [a1] M.S. Baouendi, H. Jacobowitz, F. Trèves, "On the analyticity of CR mappings" Ann. of Math. , 122 (1985) pp. 365–400 [a2] St. Bell, "Biholomorphic mappings and the $\partial$ problem" Ann. of Math. , 114 (1981) pp. 103–113 [a3] C. Fefferman, "The Bergman kernel and biholomorphic mappings of pseudoconvex domains" Inv. Math. , 26 (1974) pp. 1–65 [a4] S.G. Krantz, "Function theory of several complex variables" , Wiley (1982) pp. Chapt. 10 [a5] W. Rudin, "Function theory in the unit ball in $\mathbf C^n$" , Springer (1980) [a6] W. Rudin, "Function theory in polydiscs" , Benjamin (1969) [a7] Ph.A. Griffiths, "Entire holomorphic mappings in one and several variables" , Annals Math. Studies , 85 , Princeton Univ. Press (1976)